Thermodynamic Properties

Appendix A Thermodynamic Properties

Water

Molecular mass: M = 39.948 g/mol −3 Critical state parameters: pc = 221.2 bar, Tc = 647.3K,ρc = 17.54 × 10 mol/cm3 [1] Equilibrium liquid mass density [1]:

ρl = . + . 0.33 + . 3 mass 0 08 tanh y 0 7415 x 0 32 g/cm T x = 1 − , y = (T − 225)/46.2 Tc

Saturation vapor pressure [1]:

psat = exp [77.3491 − 7235.42465/T − 8.2lnT + 0.0057113 T ] Pa

Surface tension [1]:

−3 −3 2 γ∞ = 93.6635 + 9.133 × 10 T − 0.275 × 10 T mN/m

Lennard-Jones interaction parameters [2]:

σLJ = 2.641Å,εLJ/kB = 809.1K

Second virial coefficient [3]:

2 −3 3 B2(T ) = 17.1−102.9/(1−x) −33.6×10 (1−x) exp [5.255/(1 − x)] , cm /mol

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 293 DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013 294 Appendix A: Thermodynamic Properties

Nitrogen

Molecular mass: M = 28.0135 g/mol 3 Critical state parameters: pc = 33.958 bar, Tc = 126.192 K, ρc = 0.3133 g/cm [4] Equilibrium liquid mass density [4]:

l ρ . / ln sat = 1.48654237 x0 3294 − 0.280476066 x4 6 ρc / / + 0.0894143085 x16 6 − 0.119879866 x35 6, T x = 1 − Tc

Saturation vapor pressure [4]: psat Tc / / ln = −6.12445284 x + 1.2632722 x3 2 − 0.765910082 x5 2 − 1.77570564 x5 pc T

Surface tension [4]: 1.259 γ∞ = 29.324108 x mN/m

Lennard-Jones interaction parameters [2]:

σLJ = 3.798 Å,εLJ/kB = 71.4K

Pitzer’s acentric factor: ωP = 0.037 [2].

Mercury

Molecular mass: M = 200.61 g/mol −3 3 Critical state parameters: pc = 1510 bar, Tc = 1765 K, ρc = 23.41 × 10 mol/cm [5]. Saturation vapor pressure [5] a log p [Torr]=− + b + c lg T 10 sat T 10 with a = 3332.7, b = 10.5457, c =−0.848

Equilibrium liquid mass density [5]

ρl [ / 3]= . [ − −6( . + . 2 mass g cm 13 595 1 10 181 456 TCels 0 009205 TCels + . 3 + . 4 )] 0 000006608 TCels 0 000000067320 TCels Appendix A: Thermodynamic Properties 295 where TCels = T − 273.15 is the Celsius temperature. Surface tension [6] γ∞[mN/m]=479.4 − 0.22 TCels

Second virial coefficient [7]: TB ε1 ε2 B2 NA = c1 exp − 1 + c2 exp − − 1 T TB TB ε ε − c exp 1 − 1 + c exp − 2 − 1 cm3/mol (A.1) 1 T 2 T

3 where TB = 4286 K is the Boyle temperature of mercury, c1 = 69.87 cm /mol, 3 c2 = 22.425 cm /mol, ε1 = 655.8K,ε2 = 7563 K.

The value of coordination number N1 can be obtained from the measurements of the static structure factor. For fluid mercury it was studied over the whole liquid- vapor density range by Tamura and Hosokawa [8] and Hong et al. [9]usingX-ray diffraction measurements. Their data show that the first peak of the pair correlation function g(r) in the liquid phase is located at ≈ 3 Å and is relatively insensitive to the mass density in the range 10-13 g/cm3. The packing fraction in the liquid mercury at this range of densities is η ≈ 0.581. Using (7.68) one finds N1 ≈ 6.7. Argon

Molecular mass: M = 39.948 g/mol −3 Critical state parameters: pc = 48.6 bar, Tc = 150.633 K, ρc = 13.29 × 10 mol/cm3 [4] Equilibrium liquid mass density [4]: ρl = . + . 0.35 + . × 3 3, = − T mass M 13 290 24 49248 x 8 155083 x 10 g/cm x 1 Tc

Saturation vapor pressure [4]: p T . ln sat = c −5.904188529 x + 1.125495907 x1 5 − 0.7632579126 x3 − 1.697334376 x6 pc T

Surface tension [4]: 1.277 γ∞ = 37.78 x mN/m

Lennard-Jones interaction parameters [10]:

σLJ = 3.405 Å,εLJ/kB = 119.8K 296 Appendix A: Thermodynamic Properties

Pitzer’s acentric factor: ωP =−0.002 [2]. The second virial coefficient is given by the Tsanopoulos correlation for nonpolar substances Eq. (F.2).

N-nonane

Molecular mass: M = 128.259 g/mol −3 Critical state parameters: pc = 22.90 bar, Tc = 594.6K,ρc = 1.824 × 10 mol/cm3 [2] Equilibrium liquid mass density [2]:

ρl = . − . × −4 − . × −8 2 − . × −9 3 3 mass 0 733503 7 87562 10 TCels 9 68937 10 TCels 1 29616 10 TCels g/cm

where TCels = T − 273.15. Saturation vapor pressure [3]: −2 2 psat = exp −17.56832 ln T + 1.52556 10 T − 9467.4/T + 135.974 dyne/cm

Surface tension [3]:

γ∞ = 24.72 − 0.09347 TCels mN/m

The second virial coefficient [3]:

= . − . / + . / 2 − . / 3 − . / 8 3/ B2 NA 369 2 705 3 Tr 17 9 Tr 427 0 Tr 8 9 Tr cm mol where Tr = T/Tc. Appendix B Size of a Chain-Like Molecule

As one of the input parameters MKNT and CGNT use the size of the molecule. For a chain-like molecule, like nonane, it can be characterized by the radius of gyration Rg—the quantity used in polymer physics representing the mean square length between all pairs of segments in the chain [11]:

Nsegm 2 1 2 R = (Ri − R j )  g 2N 2 segm i, j=1 where Nsegm is the number of segments. Equivalently Rg can be rewritten as

N 1 segm R2 = (R − R )2 g N i 0 segm i=1 where R0 is the position of the center of mass of the chain. The latter expression shows that the chain-like molecule can be appropriately represented as a sphere with the radius Rg. The radius of gyration can be found using the Statistical Associating Fluid Theory (SAFT) [12]. Within the SAFT a molecule of a pure n-alkane can be modelled as a homonuclear chain with Nsegm segments of equal diameter σs and the same dispersive energy ε, bonded tangentially to form the chain. The soft-SAFT correlations for pure alkanes read [13]:

Nsegm = 0.0255 M + 0.628 (B.1) σ 3 = . + . Nsegm segm 1 73 M 22 8(B.2) where M is the molecular weight (in g/mol). Thus, the number of segments and the size of a single segment depend only on the molecular weight. Note, that within this approach Nsegm is not necessarily an integer number. Having determined Nsegm,the radius of gyration can be calculated using the Gaussian chain model in the theory of

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 297 DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013 298 Appendix B: Size of a Chain-Like Molecule polymers [11]: Nsegm Rg = σ (B.3) segm 6

Then, the effective diameter of the molecule can be estimated as

σ = 2 Rg (B.4)

For n-nonane Eqs. (B.1)–(B.2)give:

Rg = 3.202 Å,σ= 6.404 Å (B.5) Appendix C Spinodal Supersaturation for van der Waals Fluid

∗ v ∗ ∗ v In reduced units ρ = ρ /ρc, T = T/Tc, p = p /pc the van der Waals equation of state reads [14] ∗ ∗ ∗ ∗2 8ρ T p =−3ρ + (C.1) 3 − ρ∗

The spinodal equation ∂p∗/∂ρ∗ = 0is:

∗ ∗ ρ ∗ T = (3 − ρ )2 (C.2) 4 ρ∗v Solving Eq. (C.2) for the spinodal vapor density sp we obtain using the standard methods [15]: ∗v ∗ 1 ∗ ρ (T ) = 2 − 2 cos β ,β= arccos(1 − 2T ) (C.3) sp 3

Substitution of (C.3) into the van der Waals equation (C.1) yields the vapor pressure at the spinodal: ∗ 1 2 2 1 ∗v 8 4T − 3 cos β + 3 cos β sin β p = 3 3 6 (C.4) sp + 1 β 1 2 cos 3 from which the supersaturation at spinodal is

∗v ∗ p (T ) 1 S (T ∗) = sp = sp p∗ (T ∗) p∗ (T ∗) sat sat 8 4T ∗ − 3 cos 1 β + 3 cos 2 β sin2 1 β 3 3 6 (C.5) + 1 β 1 2 cos 3

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 299 DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013 Appendix D Partial Molecular Volumes

D.1 General Form

In this section we present a general framework for calculation of partial molecular volumes of components in a mixture. These quantities for a liquid phase are involved in the Kelvin equations. We consider here a general case of a two-phase m-component α, = ,..., ; α = , mixture vi i 1 m v l. The partial molecular volume of component i in the phase α is defined as ∂ α α = V , = , ,..., α = , vi α i 1 2 v l(D.1) ∂ N α, , α i p T N j, j=i where the quantities with the superscript α refer to the phase α. Since

N α V α = ρα

α the change of the total volume due to the change of Ni is α α 1 α N α dV = dN − dρ ρα i (ρα)2

α = α where we took into account that dN dNi . Then ⎡ ⎤ ∂ ρα α = 1 ⎣ − α ln ⎦ vi α 1 N α (D.2) ρ ∂ N α, , α i p T N j, j=i

The density ρα is an intensive property which can be expressed as a function of the set of intensive quantities—molar fractions of components in the phase α

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 301 DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013 302 Appendix D: Partial Molecular Volumes

N α α = j y j α (D.3) k Nk m α = ρα − α In view of normalization k=1 yk 1, is a function of m 1 variables yk and one is free to choose a particular component to be excluded from the list of independent variables. Discussing the partial molecular volume of component i, it is convenient to exclude this component from the list, i.e. to set

ρα = ρα( α,..., α , α ,..., ) y1 yi−1 yi+1 ym

Then the right-hand side of (D.2) can be expressed using the chain rule: α α ∂ α ∂ ln ρ ∂ ln ρ y j α = α α (D.4) ∂ N α, , α ∂y ∂ N i p T N j, j=i j=i j i

From (D.3) we find α α ∂y y j =− j ∂ α α (D.5) Ni v N N j, j=i

Substituting (D.5)into((D.2) and D.4) we obtain the general result α α 1 α α α ∂ ln ρ v = η ,η= 1 + y ,α= v, l(D.6) i ρα i i j ∂yα j=i j pα,T

In particular, for the binary a − b mixture (ya + yb = 1) we have: ∂ ρα ηα = − α ln a 1 yb α (D.7) ∂y α a p ,T ∂ ln ρα ηα = + yα b 1 a ∂ α (D.8) ya pα,T

Mention a useful identity resulting from (D.7)–(D.8):

ηα + ηα = ya a yb b 1(D.9)

D.2 Binary van der Waals Fluids

Here we present calculation of the partial molecular volumes of components in binary mixtures (vapor or liquid) described by the van der Waals equation of state: Appendix D: Partial Molecular Volumes 303 ρ kBT 2 p = − amρ (D.10) 1 − bmρ where the van der Waals parameters am, bm for the mixture read [2]: √ √ 2 am = (ya aa + yb ab) (D.11) bm = yaba + ybbb (D.12) where yi is the molar fraction of component i in vapor or liquid; ai and bi are the van der Waals parameters for the pure fluid i. According to the definition of the partial molecular volume consider a small perturbation in a number of molecules of component a at a fixed pressure, temperature and the number of molecules of component b. This perturbation results in the change of ρ, am and bm:

am = am + Δam (D.13) bm = bm + Δbm (D.14) ρ = ρ + Δρ (D.15)

Substituting (D.13)–(D.15) into the van der Waals Eq. (D.10) and linearizing in Δam,Δbm and Δρ we find

ρk T Δρ k T ρ k T p − B + a ρ2 = B + B (Δb ρ + Δρ b ) − ρ2Δa − 2a ρΔρ m 2 m m m m 1 − bmρ 1 − bmρ (1 − bmρ)

The left-hand side vanishes in view of (D.10) resulting in Δb k T ρ2 (1 − b ρ)2 Δρ = A Δa − m B , A ≡ m 0 m 2 0 2 (D.16) (1 − bmρ) kBT − 2amρ(1 − bmρ)

Van der Waals parameters am and bm change due to the variation in molar fractions satisfying Δya =−Δyb: √ √ √ Δam = 2 Δya am ( aa − ab) (D.17) Δbm = Δya(ba − bb) (D.18)

Substituting (D.18)into(D.16) we obtain: √ √ √ (b − b ) k T Δρ = Δy A a ( a − a ) − a b B a 0 2 m a b 2 (D.19) (1 − bmρ)

Thus, for the binary van der Waals system 304 Appendix D: Partial Molecular Volumes

Table D.1 Reduced partial pv(bar) ηl ηl molecular volumes in the a b ηl , = , 1 1.005 0.289 liquid phase, i i a b,for the mixture n-nonane 10 1.057 0.305 (a)/methane (b) at T = 240 K 25 1.142 0.289 and various total pressures 40 1.228 0.365

√ √ √ ∂ ρ vdW ( − ρ)2 ρ a ( a − a ) ρ( − ) ln = 1 bm 2 m a b − ba bb ∂ 2am ρ 2 ( − ρ)2 ya p,T 1 − (1 − b ρ) kBT 1 bm kBT m (D.20) Substituting this result into (D.7)–(D.8) we obtain the expression for ηi and hence for the partial molecular volumes 1 ∂ ln ρ vdW v = − y a ρ 1 b ∂ (D.21) ya p,T 1 ∂ ln ρ vdW v = + y b ρ 1 a ∂ (D.22) ya p,T

The reduced partial molecular volumes of the components in the liquid phase, ηl, = , i i a b play an important role for binary nucleation, especially at high pressures. TableD.1 shows the values of these parameters for the mixture n-nonane (a)/methane (b) for the nucleation temperature T = 240 K and various total pressures. Appendix E Mixtures of Hard Spheres

This appendix summarizes the relations describing the thermodynamic properties of a binary mixture of hard spheres [16–18]. The hard-sphere diameters of the components are d1 and d2; their respective number densities are ρ1 and ρ2. We start with defining the first three “moments” of the hard-sphere diameters:

Ri = di /2 = π 2 Ai di = π 3/ Vi di 6

Note that Ai can be regarded as a molecular surface area, whereas Vi denotes the molecular volume of component i. Next, the parameters ξ (k) are defined as

(0) ξ = ρ1 + ρ2 (1) ξ = ρ1 R1 + ρ2 R2 (2) ξ = ρ1 A1 + ρ2 A2 (3) ξ = ρ1V1 + ρ2V2.

Note that ξ (3) is the total volume fraction occupied by hard spheres. For notational convenience, we also introduce

( ) η = 1 − ξ 3 .

On the basis of ξ (k), new parameters c(k) are calculated according to

( ) c 0 =−ln η ξ (2) c(1) = η

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 305 DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013 306 Appendix E: Mixtures of Hard Spheres ξ (1) ξ (2) 2 c(2) = + η 8πη2 ξ (0) ξ (1)ξ (2) ξ (2) 3 c(3) = + + . η η2 12πη3

The pressure pd of the hard-sphere mixture follows from

( ) p = c 3 k T (E.1) 3 B ξ (2) 3 ξ (3) p = p − kB T (E.2) 2 3 12πη3 2p3 + p2 pd = . (E.3) 3 Let us introduce for brevity of notations two additional quantities: 3 (3) ( ) 1 − ξ ln η Y 1 = 3 2 + η2 ξ (3) ( ) 2 ξ (2) η + 1 − 2ξ 3 2lnη Y (2) = + . 6ξ (3) η3 ξ (3)

The chemical potentials μd,i can be derived from the virial equation using standard thermodynamic relationships:

(3) (0) (1) (2) (3) μ = c + c Ri + c Ai + c Vi (E.4) i ξ (2) 2 (2) (3) Ri (1) (2) μ = μ + Y − 2Ri Y (E.5) i i 3ξ (3) ( ) ( ) 2μ 3 + μ 2 μex = i i i kBT (E.6) 3 μ = ρ Λ3 + μex d,i kB T ln i i i (E.7)

The last expression presents the chemical potential of a species as a sum of an ideal and excess contributions. Appendix F Second Virial Coefficient for Pure Substances and Mixtures

Calculation of the second virial coefficient 1 −β ( ) B (T ) = 1 − e u r dr (F.1) 2 2 from first principles requires the knowledge of the intermolecular potential u(r) which is in most cases is not available. With this limited ability second virial coefficient is calculated from appropriate corresponding states correlations. For nonpolar substances such correlation has the form due to Tsanopoulos [2, 19]:

B2 pc = f0 + ωP f1 (F.2) kBTc where

= . − . / − . / 2 − . / 3 − . / 8 f0 0 1445 0 330 Tr 0 1385 Tr 0 0121 Tr 0 000607 Tr (F.3) = . + . / 2 − . / 3 − . / 8 f1 0 0637 0 331 Tr 0 423 Tr 0 008 Tr (F.4)

Tr = T/Tc is the reduced temperature and ωP is Pitzer’s acentric factor.

For normal fluids van Ness and Abbott [20] suggested simpler expressions for f0 and f1

= . − . / 1.6 f0 0 083 0 422 Tr (F.5) = . − . / 4.2 f1 0 139 0 172 Tr (F.6)

Expressions (F.5)–(F.6) agree with (F.3)–(F.4) to within 0.01 for Tr > 0.6 and ωP < 0.4, while for lower Tr the difference grows rapidly. For the mixtures the second virial coefficient is written using the mixing rule

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 307 DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013 308 Appendix F: Second Virial Coefficient for Pure Substances and Mixtures B2 = yi y j B2,ij (F.7) i j where B2,ii are the second virial coefficient of the pure components. For the cross term B2,ij the combining rules should be devised to obtain Tc,ij, pc,ij and ωP,ij which then are substituted into the pure component expression (F.2 ), with the coefficients f0 and f1 satisfying (F.3)–(F.4)or(F.5)–(F.6). For typical applications the following combining rules are used [2]

1/2 Tc,ij = (Tc,i Tc, j ) (F.8) 1/3 + 1/3 3 Vc,i Vc, j Vc,ij = (F.9) 2

Zc,i + Zc, j Zc,ij = (F.10) 2 ωP,i + ωP, j ωP,ij = (F.11) 2 Zc,ij kBTc,ij pc,ij = (F.12) Vc,ij where Vc,i = 1/ρc,i . Appendix G Saddle Point Calculations

In Chap. 13 we search for the saddle-point of the free energy of cluster formation in the space of total numbers of molecules of each species in the cluster. For calculation ( , ) l of g na nb we choose an arbitrary bulk composition ni and find the excess numbers exc = ni according to Eqs. (11.84)–(11.85). The total numbers of molecules are: ni l + exc ( , ) ni ni . Then, g na nb is found from Eq. (13.91). Is easy to see that although ( l , l ) we span the entire space of (nonnegative) bulk numbers na nb , the space of total numbers (na, nb) contains “holes”, i.e. the points, to which no value of g(na, nb) is assigned. This feature complicates the search of the saddle point of g(na, nb). To overcome this difficulty we apply a smoothing procedure aimed at elimination of the holes in (na, nb)-space by an appropriate interpolation procedure between the known values. The simplest procedure for the 2D space is the bilinear interpolation which presents the function g at an arbitrary point (na, nb) as

g(na, nb) = ana + bnb + cnanb + d (G.1) where coefficients a, b, c, d are defined by the known values of g around the point (na, nb). However, due to randomness of the location of the “holes”, the straightfor- ward application of bilinear interpolation is quite complicated. This difficulty can be avoided if we notice that (G.1) is the solution of the 2D Laplace equation

∂2 ∂2 Δg(n , n ) = 0,Δ≡ + (G.2) a b ∂ 2 ∂ 2 na nb

Thus, filling the holes in (na, nb) space by bilinear interpolation is equivalent to solving the Laplace equation (G.2), which turns out to be a quick and efficient procedure. Discretizing (G.2) on the 2D grid with the grid-size δna = δnb = 1, we have

V. I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics 860, 309 DOI: 10.1007/978-90-481-3643-8, © Springer Science+Business Media Dordrecht 2013 310 Appendix G: Saddle Point Calculations

∂2g = g(n − 1, n ) − 2g(n , n ) + g(n + 1, n ) ∂ 2 a b a b a b na ∂2g = g(n , n − 1) − 2g(n , n ) + g(n , n + 1) ∂ 2 a b a b a b nb

The discrete version of Eq. (G.2) becomes

g(na − 1, nb) + g(na, nb − 1) + g(na + 1, nb) + g(na, nb + 1) g(na, nb) = 4 (G.3) An iterative procedure of finding g(na, nb) satisfying Eq. (G.3) is known as “Laplacian smoothing” [21]. Among various possibilities of performing iterations the Gauss-seidel relaxation scheme [22] seems most computationally efficient:

i+1( − , ) + i+1( , − ) + i ( + , ) + i ( , + ) i+1 g na 1 nb g na nb 1 g na 1 nb g na nb 1 g (na, n ) = b 4 (G.4) where gi is the value of g at i-th iteration step. The procedure is repeated until i+1 i g (na, nb) ≈ g (na, nb). The saddle point of the smoothed Gibbs function satisfies ∂g ∂g = = 0 ∂n ∂n a nb b na

Note, that computationally it is preferable to search for the saddle point by solving the equivalent variational problem: 2 2 ∂g ∂g + → min (G.5) ∂n ∂n a nb b na

References

1. J. Wölk, R. Strey, J. Phys. Chem. B 105, 11683 (2001) 2. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, 4th edn. (McGraw- Hill, New York, 1987) 3. A. Dillmann, G.E.A. Meier, J. Chem. Phys. 94, 3872 (1991) 4. K. Iland, Ph.D. Thesis, University of Cologne, 2004 5. International critical tables of numerical data, vol. 2, p. 457 (McGraw-Hill, New York,1927) 6. L.E. Murr, Interfacial Phenomena in Metals and Alloys (Addison-Wisley, London, 1975) 7. A. Kaplun, A. Meshalkin, High Temp. High Press. 31, 253 (1999) 8. K. Tamura, S. Hosogawa, J. Phys. Condens. Matter 6, A241 (1994) 9. H. Hong et al., J. Non-Cryst. Solids 312—314, 284 (2002) 10. A. Michels et al., Physica 15, 627 (1949) 11. M. Sano, S.F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1988) Appendix G: Saddle Point Calculations 311

12. W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, Fluid Phase Equilib. 52, 31 (1989) 13. J. Pamies, Ph.D. Thesis, Universitat Rovira i Vrgili, Tarragona, 2003 14. V.I. Kalikmanov, Statistical Physics of Fluids. Basic Concepts and Applications (Springer, Berlin, 2001) 15. G.A. Korn, T.M. Korn, Mathematical Handbook (McGraw-Hill, New York, 1968) 16. G.A. Mansoori, N.F. Carnahan, K.E. Starling, T.W. Leland, J. Chem. Phys. 54, 1523 (1971) 17. Y. Rosenfeld, J. Chem. Phys. 89, 4272 (1988) 18. C.C.M. Luijten, Ph.D. Thesis, Eindhoven University, 1999 19. C. Tsanopoulos, AICHE J. 20, 263 (1974) 20. H.C. van Ness, M.M. Abbott, Classical Thermodynamics of Non-electrolyte Solutions (McGraw, New York, 1982) 21. F. O’Sullivan, J. Amer. Stat. Assoc. 85, 213 (1990) 22. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer, New York, 1993) Index

i, v-cluster, 71 equilibrium, 33, 80, 110, 192, 243 nonequilibrium, 24 Coarse-grained A configuration integral, 227, 232 Absorption nucleation theory(CGNT), 233 frequency, 260 Coarse-graining, 215, 228, 234 Activation Coexistence barrier, 141 pressure, 190, 240 energy for diffusion, 260 Compensation pressure Activity effect, 200, 201, 236 coefficients, 188, 193 Compressibility factor, 18, 30, 91, 104, 191 gas-phase, 192 critical, 91 liquid-phase, 194 vapor, 89 Adiabatic expansion, 277, 278, 286 Condensation nuclei Adiabatic system, 120, 121 in heterogeneous nucleation, 251 Aerosol particles Configuration integral, 84, 224 concentration, 271 Constant Angle Mie Scattering (CAMS), 279, 284 Constrained equilibrium, 25, 26, 173, 174, B 216–219, 232, 244 Barrier Contact nucleation, 55, 56, 67, 145, 150, 151, 162, angle, 141, 252, 254–256, 258, 260, 164, 258 262–265, 267–271 Binary interaction parameter, 200, 209, 210 line, 252, 255, 256, 261–264, 267 Binding energy, 84 Continuity equation, 134, 177 Binodal, 146 Cooling rate, 278, 286 Coordination number, 92, 95, 97, 129, 154, 162, 166, 229, 230 C Core-shell structure, 236 Capillarity approximat, 21, 28, 30, 32, 55, 72, Courtney correction, 32 77, 154, 182, 215, 230, 231, 252 Critical cluster, 23, 27, 28, 30, 32, 36, 37, Cluster 43–45, 47–51, 55, 73, 74, 77, distribution function, 173 97–99, 103, 136, 137, 138, 145, growth law, 34, 35, 37 150–155, 158, 174, 181, 183, 185, Cluster definition 186, 196, 198, 202, 211, 215, 220, live-time criterion, 128 233–235, 241, 257, 258, 265, 270 Cluster distribution function Cut-off radius, 164

V.I. Kalikmanov, Nucleation Theory, Lecture Notes in Physics, 860, 313 DOI: 10.1007/978-90-481-3643-8, Ó Springer Science+Business Media Dordrecht 2013 314 Index

D I Detailed balance, 25, 31, 76, 173, 217 Ideal gas, 7, 18, 28, 29, 72, 73, 75, 90, 173, Diagrammatic expansion, 225 190, 202, 216, 226, 240 Diffusion cloud chamber, 102 Ideal mixture, 190, 194, 209 Direction of principal growth approximation, Impingement rate, 25, 173, 180, 235, 243, 260 177, 220, 241, 242 average, in binary nucleation, 221, 235, Distribution function 249 one-particle, 57 Importance sampling Dupre-Young equation, 254, 262 in MC simulations, 126 Intrinsic chemical potential, 60 free energy, 57, 58, 61, 63, 65, 67 E Ising model, 268 Entropy Isothermal compressibility, 148 configurational, 84 bulk per molecule, 85 Exclusion volume, 130 K Expansion cloud chamber, 275, 277, 278 Kelvin equation classical, 30

F Fisher droplet model, 79 L Fletcher theory, 251 Lagrange equation, 115 Fluctuation theory, 19, 88, 174, 218 Landau Fokker-Planck equation, 35 expansion, 146 Frenkel distribution, 32 Laplace equation Fugacity, 82, 87, 88, 224, 228 generalized, 12 Functional standard, 12 grand potential, 65, 67, 205, 207 Latent heat, 7, 51, 121, 122, 141, 165, 199, 216 Helmholtz free energy, 55, 57–59, 65, 69, Lattice-gas model, 142 149, 205, 206 Laval Supersonic Nozzle, 162, 275, 286 Law of mass action, 32 Lennard-Jones potential, 141 Limiting consistency, 33 G Line tension, 261–265, 267–273 Gibbs Local density approximation (LDA), 62, 206 dividing surface, 9–13, 21, 44, 47, 51, 56, 76, 83, 92, 181–183, 185–187, 189, 209, 226, 236, 238, 239, 252 M equimolar surface, 10, 12, 13, 21, Matrix 107, 186 inverse, 241 surface of tension, 12, 13 transposed, 241 free energy, 20 unit, 241 of droplet formation, 20 unitary, 241, 242 Mean-field approximation, 83, 86, 96 H Mean-field Kinetic Nucleation Theory, 80, 97 Hard-spheres Metastability Carnahan-Starling theory, 63, 64, 67, 96 parameter , 200, 202, 216, 235 cavity function, 96 Metastable state, 17 effective diameter, 63 Microscopic surface tension, 79, 88, 90, 106, Heat capacity, 121 112, 157, 164, 230, 232, 233 Heat of adsorption, 260 reduced, 88, 228, 230, 233 Hill equation, 51 Mie theroy, 279, 280, 282, 285 Index 315

amplitude functions, 281 Refractive index, 279, 280 extinction coefficient, 279, 280 Retrograde nucleation, 198, 199 extinction efficiency, 280, 281 Rotation Minimum image convention, 119 angle, 241, 242, 248 Mixing rule, 205, 210, 212 transformation, 240 Modified Drupe-Young equation, 263

S N Saddle point, 55, 68, 149, 150, 171, 174, Nucleation 176–180, 185, 189, 195, 205, boundary conditions, 35, 36, 63, 119, 120, 211, 220, 221, 233, 234, 127, 136, 178, 211, 246 241–243, 249 Nucleation barrier, 22, 47, 50, 76, 77, 98, 99, Saturation 130, 132, 152–155, 158, 185, 211, pressure, 6, 168, 229 215, 235, 258, 265, 271 Scattering intensity, 282, 283, 285, 288 Nucleation pulse, 277–279, 282–284, 286 Shock tube, 275, 283–285 Nucleation pulse chamber, 278, 279 Small Angle Neutron Scattering (SANS), 286 Nucleation Theorem Small Angle X-ray Scattering first, 44, 48 (SAXS), 286–288 pressure, 53 Spherical particle second, 50 contrast factor of, 288 form-factor of, 288 Spinodal, 56, 69, 132, 145, 146, 150, 151–154, O 162–163 Order parameter, 17, 146, 149, 152 decomposition, 145, 147, 153 kinetic, 153 thermodynamic, 145, 149, 162–164 P Supercritical solution Packing fraction, 95, 97 rapid expansion of (RESS method), 124 Partial Supersaturation, 18, 151 molecular volume, 52, 182, 200, 231, 238 Surface diffusion, 259 vapor pressure, 190 Surface enrichment, 141, 181, 194, 202, 205, Partition function 207, 209, 210, 213 canonical, 58 Surface tension grand, 60 macroscopic, 69, 72, 100, 103, 104, 164, Periodic boundary conditions, 120, 127 186, 192, 199, 200, 215 Phase transition first order, 1, 7 Poisson law, 279, 284 T Pseudospinodal, 98, 145, 152–158, 162, Thermal diffusion cloud chamber, 275, 276 163, 168 Thermodynamics first law, 7 Threshold method R in MD simulations, 134–136, 138, Radius of gyration 141, 165 of a polymer molecule, 295 Time-lag, 39 Random number, 126 Tolman equation, 108 Random phase approximation (RPA), 63, 67 Tolman length, 13, 108 Random processes theory of, 272 Rate U forward, 24–26, 31, 80 Umbrella sampling, 127, 133 316 Index

V Virtual monomer approximation, 234 Van Laar Volume term, 225–227 constants, 194 model, 193 Vapor depletion, 278 W Variational transition state theory, 76 Weeks-Chandler-Anderson theory Velocity scaling decomposition scheme, 96 algorithm in MD simulations, 121 Velocity-Verlet algorithm, 118 Verlet algorithm, 118 Z Virial coefficient Zeldovich factor, 27, 98 second, 90, 106, 157, 166, 201, 229, Zeldovich relation, 37 230, 233